Signals and Systems
March 25, 2013
Summary thus far: software engineering
Focused on abstraction and modularity in software engineering.
Topics: procedures, data structures, objects, state machines
Lab Exercises: implementing robot controllers as state machines
SensorInput
Brain
Action
Abstraction and Modularity: Combinators
Cascade: make new SM by cascading two SM's
Parallel: make new SM by running two SM's in parallel
Feedback: make new SM by connecting an output to one of the inputs
Themes: PCAP
Primitives – Combination – Abstraction – Patterns
Next: signals and systems
Focus next on analysis of feedback and control systems.
Topics: difference equations, system functions, controllers.
Lab exercises: robotic steering
Understanding systems
Good news: We can build interestingly complex state machines to
control our robots.
Bad news:
• We can't predict how well our controllers are going to work, except
by running them, possibly several times, and gathering data.
• When they don't work well, we don't have any systematic way of
changing them to make them work better.
Solution:
• Make models of the controller and of the robot and its world.
• Analyze the models mathematically to characterize performance
and understand how to improve it.
State machines as models
• Make a state machine model of the plant: that is, the aspects of the
external world that you are trying to control
• Make a state machine model of your controller
• Connect the state machines (cascade and feedback)
• Run it to see what happens
Computer programs are unpredictable
Could we figure out what will happen without running the simulation,
just by looking at the definitions of the controller and the plant?
In general, no.
It is impossible to predict even whether a general computer program
will terminate and produce a result.
LTI systems are predictable
Consider simpler class of state machines:
• State: last j inputs to the system, plus last k outputs of the system
• Output: a fixed linear function of the input and the state
Linear time-invariant (LTI) systems:
• Can be analyzed mathematically, to predict behavior without
simulation
• Are compositional: cascade, parallel, and feedback compositions of
LTI systems yield LTI systems
The signals and systems abstraction
Describe a system by the way it transforms inputs into outputs.
Analyzing (and predicting) behavior
Example: use sonar sensors (i.e., currentDistance) to move robot
desiredDistance from wall.
Check yourself
Performance analysis
Quantify performance by characterizing input and output signals.
The signals and systems abstraction
Describe a system (physical, mathematical, or computational) by the
way it transforms an input signal into an output signal.
Signals and systems: widely applicable
Signals and systems abstraction has broad application: electrical,
mechanical, optical, acoustic, biological, financial, ...
Signals and systems: modular
The representation does not depend upon the physical substrate.
focuses on the flow of information, abstracts away everything else
Signals and systems: hierarchical
Representations of component systems are easily combined.
Example: cascade of component systems
Component and composite systems have the same form, and are
analyzed with same methods.
The signals and systems abstraction
Our goal is to develop representations for systems that facilitate
analysis.
Examples:
• Does the output signal overshoot? If so, how much?
• How long does it take for the output signal to reach its final value?
Continuous and discrete time
Inputs and outputs of systems can be functions of continuous time
We will focus on discrete-time systems.
Linear time-invariant systems
• linear: dependence of output on inputs is linear
• time-invariant: the same relationship between inputs and outputs
holds for any value of n
• causal: sample at time n only depends on values at the same or
previous time steps
Any LTI system can be described using a difference equation:
Feed-forward systems
Difference equation denes the output of a system at a particular time
in terms of its previous inputs
Difference Equations
Difference equations are mathematically precise and compact.
Example:
We will use the unit sample as a “ primitive“ (building-block signal) to
construct more complex signals.
Step-by-step solutions
Difference equations are convenient for step-by-step analysis.
Block diagrams
Block diagrams are useful alternative representations that highlight
visual/graphical patterns.
Same input-output behavior, different strengths/weaknesses:
• difference equations are mathematically compact
• block diagrams illustrate signal flow paths
From Samples to Signals
Operators manipulate signals rather than individual samples.
Nodes represent whole signals (e.g., X and Y ).
The boxes operate on those signals:
Signals are the primitives.
Operators are the means of combination.
Signals
A signal is an infinite sequence of sample values at discrete time steps.
Systems transduce input signals into output signals.
Operations on signals
Operators manipulate signals rather than individual samples.
Wires represent whole signals (e.g., X and Y ).
The boxes operate on those signals
Unit sample signal
Only crucial primitive in our PCAP system:
Other useful primitives are step and sinusoid signals. Discussed in
readings and exercises.
Operations on signals: scaling
Constant c often called a gain.
Operations on signals: delay
Shift signal X to the right (later in time), getting RX:
Operations on signals: addition
Add signals X1 and X2 together to get a new signal X1 + X2:
Abstracting signals
Scaling, delay, addition all return new signals that can be further
combined
Abstract by naming
Any signal with finitely many non-zero samples can be
constructed from  with delay, adder, and gain operations.
Operator notation
Symbols can now compactly represent diagrams.
Representing the difference machine
Cohort Exercise 1
Operator algebra: commutativity
Expressions involving R obey many familiar laws of algebra, e.g.,
commutativity.
R(1 - R)X = (1 - R)RX
This is easily proved by the definition of R, and it implies that cascaded
systems commute (assuming initial rest)
Operator algebra: distributivity
Multiplication distributes over addition.
Equivalent systems
Operator algebra: associativity
The associative property similarly holds for operator expressions.
Equivalent systems
Cohort Exercise 2
Feedforward and feedback systems
Feedforward: output depends only on previous inputs
Recipe: output signal equals difference between input signal and rightshifted input signal.
Feedback: output depends on previous inputs and outputs
Constraints: find the signal Y such that the difference between Y and RY
is X. But how?
Example: Accumulator
Try step-by-step analysis: it always works. Start "at rest.“
Persistent response to a transient input!
Example: Accumulator
The response of the accumulator system could also be generated by a
system with infinitely many paths from input to output, each with one
unit of delay more than the previous.
Proof in readings
Check Yourself
Check Yourself
Linear time-invariant systems
Any LTI system can be described using a difference equation of the
form:
Linear time-invariant systems
Any LTI system can be described using a difference equation of the
form:
Any LTI system can be described using an operator equation of the
form:
Combining modules
Assign names to all wires
LTI systems as state machines
LTI systems as state machines
Multiple representations of LTI systems
• difference equations: good for step-by-step simulation
• block diagrams: good for signal-flow intuition
• operator expressions: good for compact description and combining
systems
• Python SM subclasses: implementation of difference equation
• Python combination of primitive SMs: implementation of
difference equation, but easier to get right
This Week
Readings: Chapter 5.1-5.4 of Digital World Notes (mandatory!)
Cohort Exercises & Homework: Practice on LTI systems (note the due
dates & times)
Cohort Session 2 & 3: Constructing and using LTI model for robot
control